vector-spaces

# Vector Spaces ## When to Use Use this skill when working on vector-spaces problems in linear algebra. ## Decision Tree 1. **Check Subspace** - Contains zero vector? - Closed under addition? - Closed under scalar multiplication? - Verify with `z3_solve.py prove` 2. **Linear Independence** - Set up Ax = 0 where columns are vectors - `sympy_compute.py nullspace "A"` - Trivial nullspace = independent 3. **Basis and Dimension** - Find spanning set, remove dependent vectors - `sympy_compute.py rref "A"` to find pivot columns - Dimension = number of pivots 4. **Change of Basis** - Find transition matrix P - New coords = P^(-1) * old coords - `sympy_compute.py inverse "P"` ## Tool Commands ### Sympy_Nullspace ```bash uv run python -m runtime.harness scripts/sympy_compute.py nullspace "[[1,2,3],[4,5,6]]" ``` ### Sympy_Rref ```bash uv run python -m runtime.harness scripts/sympy_compute.py rref "[[1,2,3],[4,5,6]]" ``` ### Z3_Prove ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "subspace_closed" ``` ## Cognitive Tools Reference See `.claude/skills/math-mode/SKILL.md` for full tool documentation.